Game by game Atlético Madrid have arrived at the game. On Saturday night, Barcelona visit the Vicente Calderón stadium, down on the banks of the Manzanares. It is week 19, marking the halfway point of the season when everyone has played everyone else, and the reward for victory is the top of the table, a significant lead going into the back 19. The country's two outstanding teams meet at last, another Madrid-Barcelona presented as a title-decider. Only this time the Madrid in question is Atlético.

For weeks, eyes have been drawn to this match; for weeks, months even, coach Diego "El Cholo" Simeone has been asked about it, about his side's chances of achieving what no one has achieved in a decade and win the title ahead of Barça and Madrid. With each passing match, the question becomes more insistent. Now can you win the league? Every time, he insists that his team are progressing partido a partido, game by game, while describing winning the league as impossible.

The cliche is not new but it has become his, almost as if Simeone invented it; game by game has become a running joke. Simeone refused to get ahead of himself and game by game has been good for his team. This was always going to be a huge match but, in truth, few expected Atlético to reach this point quite so well positioned: level on points with Barcelona, five ahead of Real Madrid: 18 games played, 16 wins, one draw and a solitary defeat which came thanks to a fluke own goal at Espanyol. They have conceded just 11 goals all season and scored 47.

"This league is boring: Madrid and Barcelona will be ahead. As games go by you'll see I'm right," Simeone said after that loss at Espanyol, repeating his mantra from the start of the campaign. But the games pass one by one, and Atlético are still there. If they win on Saturday, Simeone will be asked again: And now? The response will probably be the same, but the sensations will surely be different. Those who have resisted the belief that Atlético really can win the league may convert too.

Diego Simeone took over two days before Christmas 2011. Atlético had just been knocked out of the Copa del Rey by third-tier Albacete. Over the years, a legend had built up that had Atlético as el pupas, the jinxed one. What could go wrong would go wrong: losing the European Cup final to a fluke goal, getting relegated, always, always losing to their city rivals.

The legend served as an excuse, a convenient smokescreen for failure, and it wasn't always true: Atlético had won the Europa League in 2010. But it was true that Atlético were a club in almost permanent crisis, a state of confusion and conflict, fractured, fragile and fatalistic.

Simeone has changed everything, the club's very identity. Captain of Atlético's double-winning side in 1996, carrying a unique moral authority, he brought the club together: fans, management, board. "I'm not sure if he was the only one who could do that but he was certainly the best placed," the club captain Gabi told Panenka magazine.

Gabi, brought up in Atlético's youth system, had played with Simeone. He explains: "El Cholo had the fans on his side because of the memory he had left as a player, an idol. He knew the board because they had had him as a player. And he knew what he had in the squad. He had all the support necessary. But that was a double-edged sword, because if results had not been good he would have gone from hero to failure.

"We were sunk mentally," Gabi admits. Three-thousand fans came to Simeone's presentation. "They'll turn on me too if we lose," he told the players; they were in this together. There was talent in the team, but Simeone changed their mentality.

Few represent that change in mentality like the striker Diego Costa: he joined Atlético in 2007 but went on loan to Celta, Albacete, Valladolid, Rayo and only a serious injury prevented a permanent sale. Nurtured by Simeone, this summer for the first time he knew where he would be playing. His best ever goalscoring return in the league was 10. He has 19 already. Five of them have been provided by Koke: no combination has been as productive in the league.

"Atlético play like Diego Simeone played: tough, focused and tactically perfect," the Real Madrid coach Carlo Ancelotti said in the buildup to the city derby earlier this season. The following night, Atlético won 1-0. It was their second successive win at the Bernabéu, for the first time ever. The first, their first of any sort against Madrid in 25 games going back 14 years, had been in the final of the Copa del Rey four months before. Some considered it the biggest victory in their history.

In two years under Simeone, Atlético have won the Europa League, the European Super Cup and the Copa del Rey. They returned to the Champions League, winning five, drawing one, and conceding just three. Now they've beaten club records – and, together with Barcelona, league ones too – despite having a budget a fifth of the size of the big two. Those resources – Atlético have a shorter squad that may not assimilate injuries – could cost them but so far the collapse has stubbornly refused to come.

Barcelona are one of the few sides to resist. Three times Atlético have played them in the league under Simeone, three times they have lost. But they are improving: in August they met twice in the Spanish Super Cup and Barcelona only took the title on away goals after two draws, 1-1 at the Calderón, 0-0 at the Camp Nou. Now, Atlético stand level with them at the top, racking up 49 points. Game by game. On Saturday night, the deadlock must be broken; Saturday night is the game.